I Newton's formula for the sums of powers of roots?

Please take a look of the photo. In the middle part, it says For each I, by division and gets the following results. Please further explain to me how to get the result by division. The photo is attached.

Attempt: 1. Using f(x)=α0(x-α1)(x-α2)...(x-αn) form. If it is divided by x-αi , there should be no αi occurs in the results.
2. Using fundamental theorem of algebra. a1/a0=α1+α2+α3+α4+..+αn, αi is lost so it is (a1-a0αi)x^(n-2),not(a1+a0αi)x^(n-2), so this is not the method. Then how should i divide it? Please help